arXiv subjects
Publications and source records attributed to Stephen Landsittel.
We prove that ordinary Hilbert-Kunz multiplicity can be transcendental. More precisely, over every uncountable algebraically closed field of characteristic $p>2$, there exists a normal standard graded domain $S$ and an $S_+$-primary homogeneous ideal $I\subseteq S$ such that $e_{\rm HK}(IS_{S_+})$ is transcendental.
We give a systematic analysis of the analytic spread of the determinantal ideal $J_{G,H}$ arising from a pair of graphs $(G, H)$. We give sharp bounds for this analytic spread, and combinatorial conditions and obstructions for its maximality via a linear matroid. When $H$ is a single edge, $J_{G,H}$ is isomorphic to the binomial edge ideal $J_G$ and its analytic spread is shown to equal the rank of $G$ in Kalai's $2$-hyperconnectivity matroid.
We prove that epsilon multiplicity can take transcendental values. The main structural result is a one-ideal formula for section rings: under natural positivity hypotheses, the epsilon multiplicity of an ideal generated in one degree is equal to an integral of a divisor-volume function. This formula transports an asymptotic colength invariant of ideals to the geometry and arithmetic of divisor volumes. To produce a transcendental value, we combine the formula with a shifted projective-bundle construction inspired by Borntr\"ager and Nickel. The shift places the construction in the positivity range required by the one-ideal formula while preserving the underlying disk geometry of the volume computation. Reversing the order of integration reduces the resulting integral to three integrals of rational functions. Their arctangent terms cancel exactly, whereas the remaining real logarithms form an explicit algebraic linear combination whose value is positive. Baker's theorem then implies transcendence. Consequently, there exists a homogeneous ideal in a normal standard graded domain whose epsilon multiplicity is transcendental.
In this paper we observe when saturation of ideals in a local ring R commutes with extension along ring maps and initial ideals. We give a characterization in terms of Cohen Macaulayness for when this happens along the map from R to R modulo its nilradical. We give several examples and non examples where this happens, and we demonstrate an application of this condition to epsilon multiplicity. Additionally, we show that saturation commutes with extension along a flat injection of local rings if and only if the closed fiber of the injection is Artinian.
In this paper we give a construction for a linear quotient ordering of a class of products of two ideals which have linear quotients. We apply this construction to give a class of modified anticycle graphs whose square and cube have linear quotients.
In this paper, we initiate a systematic study of the generalized Hilbert-Kunz multiplicity for families of ideals in a Noetherian local ring (R,m) of positive characteristic, and introduce a new asymptotic invariant called the Amao-type multiplicity. We establish that, for a p-family of ideals, the generalized Hilbert-Kunz multiplicity arises as the limit of Amao-type multiplicities.
We investigate the analytic spread of binomial edge ideals of finite simple graphs. We provide tight bounds for this invariant in general. For special families of graphs (e.g., closed graphs, pseudo-forests), we compute the exact value for the analytic spread of the corresponding binomial edge ideals via combinatorial and convex geometric means.
We prove that the epsilon multiplicity exists in a Noetherian local ring whenever the nildradical of the completion of R has nonmaximal dimension. We also extend the volume equals multiplicity formula for the epsilon multiplicity to this setting.
In this paper we prove a volume = multiplicity type theorem for epsilon multiplicity. We show that epsilon multiplicity is a limit of Amao multiplicities.